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The history of mathematical notation[1] includes the commencement, progress, and cultural diffusion of mathematical symbols and the conflict the methods of notation confronted in a notation's move to popularity or inconspicuousness. Mathematical notation[2] comprises the symbols used to write mathematical equations and formulas. Notation generally implies a set of well-defined representations of quantities and symbols operators.[3] The history includes Hindu-Arabic numerals, letters from the Roman, Greek, Hebrew, and German alphabets, and a host of symbols invented by mathematicians over the past several centuries.

The development of mathematical notation can be divided in stages.[4][5] The "rhetorical" stage is where calculations are performed by words and no symbols are used.[6] The "syncopated" stage is where frequently used operations and quantities are represented by symbolic syntactical abbreviations. From ancient times through the post-classical age,[note 1] bursts of mathematical creativity were often followed by centuries of stagnation. During the Middle Ages, the worldwide spread of knowledge began, and written examples of mathematical developments came to light. The "symbolic" stage is where comprehensive systems of notation supersede rhetoric. Beginning in Islamic North Africa in the 13th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day. This symbolic system was in use by Islamic mathematicians since the 15th century, and by Indian and European mathematicians since the middle of the 17th century,[7] and has continued to develop in the contemporary era.

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, the focus here, the investigation into the mathematical methods and notation of the past.

For more details on particular notations, see: Arithmetic (History of arithmetic), Algebra (History of algebra), Geometry (History of geometry), Trigonometry (History of trigonometry), Calculus (History of calculus)

Rhetorical stage

The history of mathematical notation commences with the tally marks of prehistoric Africans, and then the investigations of the ancient Mesopotamians, Egyptians and Phoenicians in the ancient Near East. Numerical notation distinctive feature, i.e. symbols having local as well as intrinsic values (arithmetic), implies a state of civilization at the period of its invention. Our knowledge of the mathematical attainments of these early peoples, to which this section is devoted, is imperfect and the following brief notes be regarded as a summary of the conclusions which seem most probable, and the history of mathematics begins with the symbolic sections.

Many areas of mathematics began with the study of real world problems, before the underlying rules and concepts were identified and defined as abstract structures. For example, geometry has its origins in the calculation of distances and areas in the real world; algebra started with methods of solving problems in arithmetic.

There can be no doubt that most early peoples which have left records knew something of numeration and mechanics, and that a few were also acquainted with the elements of land-surveying. In particular, the Egyptians paid attention to geometry and numbers, and the Phoenicians to practical arithmetic, book-keeping, navigation, and land-surveying. The results attained by these people seem to have been accessible, under certain conditions, to travelers. It is probable that the knowledge of the Egyptians and Phoenicians was largely the result of observation and measurement, and represented the accumulated experience of many ages.

Beginning of notation

Written mathematics began with numbers expressed as tally marks, with each tally representing a single unit. For example, one notch in a bone represented one animal, or person, or anything else. This originated from prehistoric Africa.

The numerical symbols consisted probably of strokes or notches cut in wood or stone, and intelligible alike to all nations.[note 2] For example, one notch in a bone represented one animal, or person, or anything else. Greek tradition uniformly assigned the special development of geometry to the Egyptians, and that of the science of numbers[note 3] either to the Egyptians or to the Phoenicians.

Mesopotamian notation

The Mesopotamians had symbols for each power of ten.[8] Later, they wrote their numbers in almost exactly the same way done in modern times. Instead of having symbols for each power of ten, they would just put the coefficient of that number. Each digit was at separated by only a space, but by the time of Alexander, they had created a symbol that represented zero and was a placeholder. The Mesopotamians also used a sexagesimal system, that is base sixty. It is this system that is used in modern times when measuring time and angles. Babylonian mathematics is derived from more than 400 clay tablets unearthed since the 1850s.[9] Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. Some of these appear to be graded homework. The earliest evidence of written mathematics dates back to the ancient Sumerians and the system of metrology from 3000 BC. From around 2500 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.[10]

The majority of Mesopotamian clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, and the calculation of regular reciprocal pairs.[11] The tablets also include multiplication tables and methods for solving linear and quadratic equations. The Babylonian tablet YBC 7289 gives an approximation of √2 accurate to five decimal places. Babylonian mathematics were written using a sexagesimal (base-60) numeral system. From this derives the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, as well as the use of minutes and seconds of arc to denote fractions of a degree. Babylonian advances in mathematics were facilitated by the fact that 60 has many divisors: the reciprocal of any integer which is a multiple of divisors of 60 has a finite expansion in base 60. (In decimal arithmetic, only reciprocals of multiples of 2 and 5 have finite decimal expansions.) Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. They lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.

Egyptian notation

The ancient Egyptians had a symbolic notation which was the numeration by Hieroglyphics.[12][13] Ancient Egyptian mathematics had a symbol for one, ten, one-hundred, one-thousand, ten-thousand, one-hundred-thousand, and one-million. Smaller digits were placed on the left of the number, as they are in Arabic numerals. Later, the Egyptians used hieratic instead of hieroglyphic script to show numbers. Hieratic was more like cursive and replaced several groups of symbols with individual ones. For example, the four vertical lines used to represent four were replaced by a single horizontal line. This is found in the Rhind Mathematical Papyrus (c. 2000–1800 BC) and the Moscow Mathematical Papyrus (c. 1890 BC). The system the Egyptians used was discovered and modified by many other civilizations in the Mediterranean. The Egyptians also had symbols for basic operations: legs going forward represented addition, and legs walking backward to represent subtraction.

Greek notation

The ancient Greeks at first employed Attic numeration, which was based on the system of the Egyptians and was later adapted and used by the Romans. Numbers one through four were vertical lines, like in the hieroglyphics. The symbol for five was the Greek letter pente, which was the first letter of the word for five. Numbers six through nine were pente with vertical lines next to it. Ten was represented by the first letter of the word for ten, deka, one-hundred by the first letter from the word for one-hundred, etc.

The Ionian numeration used the entire alphabet and three archaic letters.

 A B Г Δ E F Z H θ I K Λ M N Ξ O Π (qoppa) P Σ T Υ Ф X Ψ Ω (sampi) 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900

This system appeared in the third century B.C., before the letters vau (F), koppa, and sampi became archaic. When lowercase letters appeared, these replaced the uppercase ones as the symbols for notation. Multiples of one-thousand were written as the first nine numbers with a stroke in front of them; thus one-thousand was, α, two-thousand was, β, etc. M was used to multiply numbers by ten-thousand. The number 88,888,888 would be written as M,ηωπη*ηωπη[14]

Greek mathematical reasoning was almost entirely geometric (albeit often used to reason about nongeometric subjecs such as number theory), and hence the Greeks had no interest in algebraic symbols. The great exception was Diophantus of Alexandria the first great algebraists. His Arithmetica was one of the first texts to use symbols in equations. It was not completely symbolic, but was much more than previous books. An unknown number was called s. The square of s was $\Delta^y$; the cube was $K^y$; the fourth power was $\Delta^y\Delta$; and the fifth power was $\Delta K^y$. The expression $2x^4+3x^3-4x^2+5x-6$ would be written as SS2 C3 x5 M S4 u6.

Chinese notation

The Chinese used numerals that look much like the tally system. Numbers one through four were horizontal lines. Five was an X between two horizontal lines; it looked almost exactly the same as the Roman numeral for ten. Nowadays, the huāmǎ system is only used for displaying prices in Chinese markets or on traditional handwritten invoices.

Syncopated stage

Indian notation

The origin of our present system of numerical notation is ancient. There is no doubt that it was in use among the Hindus over two thousand years ago.

The algebraic notation of the Indian mathematician, Brahmagupta, was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms.[15]

The Hindu-Arabic numeral system and the rules for the use of its operations, in use throughout the world today, likely evolved over the course of the first millennium AD in India and was transmitted to the west via Islamic mathematics.[16][17]

Arabic numerals and Islamic notation

Despite their name, Arabic numerals actually started in India. The reason for this misnomer is Europeans first saw the numerals used in an Arabic book, Concerning the Hindu Art of Reckoning, by Mohommed ibn-Musa al-Khwarizmi.

Al-Khwārizmī wrote several important books on the Hindu-Arabic numerals and on methods for solving equations. His book On the Calculation with Hindu Numerals, written about 825, along with the work of Al-Kindi,[note 4] were instrumental in spreading Indian mathematics and Indian numerals to the West.

Al-Khwarizmi did not claim the numerals as Arabic, but over several Latin translations, the fact that the numerals were Indian in origin was lost.

The word algorithm is derived from the Latinization of Al-Khwārizmī's name, Algoritmi, and the word algebra from the title of one of his works, Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala (The Compendious Book on Calculation by Completion and Balancing).

Islamic mathematics developed and greatly expanded the mathematics of the ancient Near Eastern, Indian and Central Asian[18] civilizations. Al-Khwārizmī gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots,[19] and Al-Khwārizmī was to teach algebra in an elementary form and for its own sake.[20] Al-Khwārizmī also discussed the fundamental method of "reduction" and "balancing", referring to the transposition of subtracted terms to the other side of an equation, that is, the cancellation of like terms on opposite sides of the equation. This is the operation which al-Khwārizmī originally described as al-jabr.[21] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study." Al-Khwārizmī also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems."[22]

Al-Karaji, in his treatise al-Fakhri, extends the methodology to incorporate integer powers and integer roots of unknown quantities.[note 5][23] The historian of mathematics, F. Woepcke,[24] praised Al-Karaji for being "the first who introduced the theory of algebraic calculus." Also in the 10th century, Abul Wafa translated the works of Diophantus into Arabic. Ibn al-Haytham would develop analytic geometry. Al-Haytham derived the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. Al-Haytham performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree.[note 6][25] In the late 11th century, Omar Khayyam would develop algebraic geometry, wrote Discussions of the Difficulties in Euclid,[note 7] and wrote on the general geometric solution to cubic equations. Nasir al-Din Tusi (Nasireddin) made advances in spherical trigonometry. Muslim mathematicians during this period include the addition of the decimal point notation to the Arabic numerals.

Many Greek and Arabic texts on mathematics were then translated into Latin, which led to further development of mathematics in medieval Europe. In the 12th century, scholars traveled to Spain and Sicily seeking scientific Arabic texts, including al-Khwārizmī's[note 8] and the complete text of Euclid's Elements.[note 9][26][27]

One of the first European books that advocated using the numerals was Liber Abaci, by Leonardo of Pisa, better known as Fibonacci. Liber Abaci is better known for the mathematical problem Fibonacci wrote in it about a population of rabbits. The growth of the population ended up being a Fibonacci sequence, where a term is the sum of the two preceding terms.

Symbolic stage

Islamic notation

Abū al-Hasan ibn Alī al-Qalasādī (1412-1482) was the last major medieval Arab algebraist, who improved on the algebraic notation earlier used in the Maghreb by Ibn al-Banna in the 13th century[28] and by Ibn al-Yāsamīn in the 12th century.[29] In contrast to the syncopated notations of their predecessors, Diophantus and Brahmagupta, which lacked symbols for mathematical operations,[30] al-Qalasadi's algebraic notation was the first to have symbols for these functions and was thus "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.[28]

In the 15th century, Ghiyath al-Kashi computed the value of π to the 16th decimal place. Kashi also had an algorithm for calculating nth roots.[note 10]

Renaissance European notation

Two of the most widely used mathematical symbols are addition and subtraction, + and −. The plus sign was first used by Nicole Oresme in Algorismus proportionum, possibly an abbreviation for "et", which is "and" in Latin (in much the same way the ampersand began as "et"). The minus sign was first used by Johannes Widmann in Mercantile Arithmetic. Widmann used the minus symbol with the plus symbol, to indicate deficit and surplus, respectively.[31] The symbol for the constant pi, π, was also first used during this time. William Jones used π in Synopsis palmariorum mathesios in 1706 because it is the first letter of the Greek word perimetron (περιμετρον), which means perimeter in Greek. Ironically, the mathematician Leonhard Euler (who would begin much of his own notation that is used today) did not use π but its equivalent in the Roman alphabet, p. However, others during Euler's time and almost all after it used Jones's notation.

Calculus notation

Calculus had two main systems of notation, each created by one of the creators: that developed by Isaac Newton and the notation developed by Gottfried Leibniz. Leibniz's is the notation used most often today. Newton's was simply a dot or dash placed above the function. For example, the derivative of the function x would be written as $\dot{x}$. The second derivative of x would be written as $\ddot{x}$, etc. In modern usage, this notation generally denotes derivatives of physical quantities with respect to time, and is used frequently in the science of mechanics.

Leibniz, on the other hand, used the letter d as a prefix to indicate differentiation, and introduced the notation representing derivatives as if they were a special type of fraction. For example, the derivative of the function x with respect to the variable t in Leibniz's notation would be written as ${ dx \over dt }$. This notation makes explicit the variable with respect to which the derivative of the function is taken.

Leibniz also created the integral symbol, $\int_{-N}^{N} e^x\, dx$. The symbol is an elongated S, representing the Latin word Summa, meaning "sum". When finding areas under curves, integration is often illustrated by dividing the area into tall, thin rectangles. Infintesimally thin rectangles, when added, yield the area. The process of add up the infintesmal areas in integration, hence the S for sum.

Euler notation

Leonhard Euler was one of the most prolific mathematicians in history, and perhaps was also the most prolific inventor of canonical notation. His contributions include his use of e to represent the base of natural logarithms. It is not known exactly why e was chosen, but it was probably because the first four letters of the alphabet were already commonly used to represent variables and other constants. Euler was also one of the first to use π to represent pi consistently. The use of π was first suggested by William Jones, who used it as shorthand for perimeter. Euler was also the first to use i to represent the square root of negative one, $\sqrt{-1}$, although he earlier used it as an infinite number. (Nowadays the symbol created by John Wallis, $\infty$, is used for infinity.) For summation, Euler was the first to use sigma, Σ, as in $\sum_{n=1}^\infty\frac{1} {n^2}$. For functions, Euler was the first to use the notation $f(x)$ to represent a function of x.

Logic notation

Once logic was recognized as an important part of mathematics, it received its own notation. Some of the first was the set of symbols used in Boolean algebra, created by George Boole in 1854. Boole himself did not see logic as a branch of mathematics, but it has come to be encompassed anyway. Symbols found in Boolean algebra include $\land$ (AND), $\lor$ (OR), and $\lnot$ (NOT). With these symbols, and letters to represent different elements, one can make logical statements such as $a\lor\lnot a=1$, that is "The existence of element a OR the existence of element NOT a is 1", meaning it is true either a exists or it doesn't. Boolean algebra has many practical uses as it is, but it also was the start of what would be a large set of symbols to be used in logic. Most of these symbols can be found in propositional calculus, a formal system described as $\mathcal{L} = \mathcal{L}\ (\Alpha,\ \Omega,\ \Zeta,\ \Iota)$. $\Alpha$ is the set of elements, such as the a in the example with Boolean algebra above. $\Omega$ is the set that contains the subsets that contain operations, such as $\lor$ or $\land$. $\Zeta$ contains the inference rules, which are the rules dictating how inferences may be logically made, and $\Iota$ contains the axioms. (See also: Basic and Derived Argument Forms). With these symbols, proofs can be made that are completely artificial.

While proving his incompleteness theorems, Kurt Gödel created an alternative to the symbols normally used in logic. He used Gödel numbers, which were numbers that represented operations with set numbers, and variables with the first prime numbers greater than 10. (See a table of the numbers here). With Gödel numbers, logic statements can be broken down into a number sequence. Gödel then took this one step farther, taking the first n prime numbers and putting them to the power of the numbers in the sequence. These numbers were then multiplied together to get the final product, giving every logic statement its own number.[32] For example, take the statement "There is exists a number x so that it is not y". Using the symbols of propositional calculus, this would become $(\exists x)(x=\lnot y)$. If the Gödel numbers replace the symbols, it becomes {8, 4, 11, 9, 8, 11, 5, 1, 13, 9}. There are ten numbers, so the first ten prime numbers are found and these are {2, 3, 5, 7, 11, 13, 17, 19, 23, 29}. Then, the Gödel numbers are made the powers of the respective primes and multiplied, giving $2^8\times3^4\times5^{11}\times7^9\times11^8\times13^{11}\times17^5\times19^1\times23^{13}\times29^9$. The resulting number is approximately $3.096262735\times10^{78}$.

Future of mathematical notation

In the history of mathematical notation, ideographic symbol notation has come full circle with the rise of computer visualization systems. The notations can be applied to abstract visualizations, such as for rendering some projections of a Calabi-Yau manifold. Examples of abstract visualization which properly belong to the mathematical imagination can be found in computer graphics. The need for such models abounds, for example, when the measures for the subject of study are actually random variables and not really ordinary mathematical functions.

Main relevance
Abuse of notation, Well-formed formula, Big O notation (L-notation), Dowker notation, Hungarian notation, Infix notation, Positional notation, Polish notation (Reverse Polish notation), Sign-value notation, Subtractive notation, infix notation
Numbers and quantities
List of numbers, Irrational and suspected irrational numbers, γ, ζ(3), 2, 3, 5, φ, ρ, δS, α, e, π, δ, Physical constants, c, ε0, h, G, Greek letters used in mathematics, science, and engineering
General relevance
Order of operations, Scientific notation (Engineering notation), Actuarial notation
Dot notation
Chemical notation (Lewis dot notation (Electron dot notation)), Dot-decimal notation
Arrow notation
Knuth's up-arrow notation, infinitary combinatorics (Arrow notation (Ramsey theory))
Geometries
Projective geometry, Affine geometry, Finite geometry
Lists and outlines
Outline of mathematics (Mathematics history topics and Mathematics topics (Mathematics categories) ), Mathematical theories ( First-order theories, Theorems and Disproved mathematical ideas), Mathematical proofs (Incomplete proofs), Mathematical identities, Mathematical series, Mathematics reference tables, Mathematical logic topics, Mathematics-based methods, Mathematical functions, Transforms and Operators, Points in mathematics, Mathematical shapes, Knots (Prime knots and Mathematical knots and links), Inequalities, Mathematical concepts named after places, Mathematical topics in classical mechanics, Mathematical topics in quantum theory, Mathematical topics in relativity, String theory topics, Unsolved problems in mathematics, Mathematical jargon, Mathematical examples, Mathematical abbreviations
Misc.
Hilbert's problems, Mathematical coincidence, Chess notation, Line notation, Musical notation (Dotted note), Whyte notation, Dice notation, recursive categorical syntax
People
Mathematicians (Amateur mathematicians and Female mathematicians), Thomas Bradwardine, Thomas Harriot, Felix Hausdorff, Gaston Julia, Helge von Koch, Paul Lévy, Aleksandr Lyapunov, Benoit Mandelbrot, Lewis Fry Richardson, Wacław Sierpiński, Saunders Mac Lane, Paul Cohen, Gottlob Frege, G. S. Carr, Robert Recorde, Bartel Leendert van der Waerden, G. H. Hardy, E. M. Wright, James R. Newman, Carl Gustav Jacob Jacobi, Roger Joseph Boscovich, Eric W. Weisstein, Mathematical probabilists, Statisticians

General
Other

Notes

1. Or the Middle Ages.
2. Such characters, in fact, are preserved with little alteration in the Roman notation, an account of which may be found in John Leslie's Philosophy of Arithmetic.
3. Number theory is branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers).
4. Al-Kindi also introduced cryptanalysis and frequency analysis.
5. Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.
6. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.
7. a book about what he perceived as flaws in Euclid's Elements, especially the parallel postulate
8. translated into Latin by Robert of Chester
9. translated in various versions by Adelard of Bath, Herman of Carinthia, and Gerard of Cremona
10. This was a special case of the methods given many centuries later by Ruffini and Horner.

References and citations

General
Citations
1. Florian Cajori. A History of Mathematical Notations: Two Volumes in One. Cosimo, Inc., Dec 1, 2011
2. A Dictionary of Science, Literature, & Art, Volume 2. Edited by William Thomas Brande, George William Cox. Pg 683
3. Notation -- from Wolfram MathWorld http://mathworld.wolfram.com/Notation.html
4. Diophantos of Alexandria: A Study in the History of Greek Algebra. By Sir Thomas Little Heath. Pg 77.
5. Mathematics: Its Power and Utility. By Karl J. Smith. Pg 86.
6. The Commercial Revolution and the Beginnings of Western Mathematics in Renaissance Florence, 1300-1500. Warren Van Egmond. 1976. Page 233.
7. Solomon Gandz. "The Sources of al-Khowarizmi's Algebra"
8. Mathematics in Egypt and Mesopotamia
9. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "Mesopotamia" p. 25.
10. Duncan J. Melville (2003). Third Millennium Chronology, Third Millennium Mathematics. St. Lawrence University.
11. Aaboe, Asger (1998). Episodes from the Early History of Mathematics. New York: Random House. pp. 30–31.
12. Encyclopædia Americana. By Thomas Gamaliel Bradford. Pg 314
13. Mathematical Excursion, Enhanced Edition: Enhanced Webassign Edition By Richard N. Aufmann, Joanne Lockwood, Richard D. Nation, Daniel K. Cleg. Pg 186
14. Boyer, Carl B. A History of Mathematics, 2nd edition, John Wiley & Sons, Inc., 1991.
15. (Boyer 1991, "China and India" p. 221) "he was the first one to give a general solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. [...] It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India - or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words."
16. Robert Kaplan, "The Nothing That Is: A Natural History of Zero", Allen Lane/The Penguin Press, London, 1999
17. "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius." - Pierre Simon Laplace http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_numerals.html
18. A.P. Juschkewitsch, "Geschichte der Mathematik im Mittelalter", Teubner, Leipzig, 1964
19. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "The Arabic Hegemony" p. 230. (cf., "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwārizmī's exposition that his readers must have had little difficulty in mastering the solutions.")
20. Gandz and Saloman (1936), The sources of Khwarizmi's algebra, Osiris i, pp. 263–77: "In a sense, Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".
21. Boyer, C. B. A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, 1989 ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7). "The Arabic Hegemony" p. 229. (cf., "It is not certain just what the terms al-jabr and muqabalah mean, but the usual interpretation is similar to that implied in the translation above. The word al-jabr presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word muqabalah is said to refer to "reduction" or "balancing" - that is, the cancellation of like terms on opposite sides of the equation.")
22. Rashed, R.; Armstrong, Angela (1994). The Development of Arabic Mathematics. Springer. pp. 11–12. ISBN 0-7923-2565-6. OCLC 29181926.
23. Victor J. Katz (1998). History of Mathematics: An Introduction, pp. 255–59. Addison-Wesley. ISBN 0-321-01618-1.
24. F. Woepcke (1853). Extrait du Fakhri, traité d'Algèbre par Abou Bekr Mohammed Ben Alhacan Alkarkhi. Paris.
25. Victor J. Katz (1995), "Ideas of Calculus in Islam and India", Mathematics Magazine 68 (3): 163–74.
26. Marie-Thérèse d'Alverny, "Translations and Translators", pp. 421–62 in Robert L. Benson and Giles Constable, Renaissance and Renewal in the Twelfth Century, (Cambridge: Harvard University Press, 1982).
27. Guy Beaujouan, "The Transformation of the Quadrivium", pp. 463–87 in Robert L. Benson and Giles Constable, Renaissance and Renewal in the Twelfth Century, (Cambridge: Harvard University Press, 1982).
28. 28.0 28.1
29. Prof. Ahmed Djebbar (June 2008). "Mathematics in the Medieval Maghrib: General Survey on Mathematical Activities in North Africa". FSTC Limited. Retrieved 2008-07-19.
30. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."
31. Miller, Jeff. "Earliest Uses of Various Mathematical Symbols." 04 June 2006. Gulf High School. 24 September 2006 <http://web.archive.org/web/20081204035420/http://members.aol.com/jeff570/mathsym.html>.
32. Casti, John L. 5 Golden Rules. New York: MJF Books, 1996.